Individual-based approaches to birth and death in avascu1ar tumors

Heterogeneous biomechanical/mathematical modeling of spatial prediction of glioblastoma progression using magnetic resonance imaging-based finite element method

BACKGROUND AND OBJECTIVE

Brain tumors are one of the most common diseases and causes of death in humans. Since the growth of brain tumors has irreparable risks for the patient, predicting the growth of the tumor and knowing its effect on the brain tissue will increase the efficiency of treatment strategies.

METHODS

This study examines brain tumor growth using mathematical modeling based on the Reaction-Diffusion equation and the biomechanical model based on continuum mechanics principles. With the help of the image threshold technique of magnetic resonance images, a heterogeneous and close-to-reality environment of the brain has been modeled and experimental data validated the results to achieve maximum accuracy in predicting growth.

RESULTS

The obtained results have been compared with the reported conventional models to evaluate the presented model. In addition to incorporating the chemotherapy effects in governing equations, the real-time finite element analysis of the stress tensors of the surrounding tissue of tumor cells and considering its role in changing the shape and growth of the tumor has added to the importance and accuracy of the current model.

CONCLUSIONS

The comparison of the obtained results with conventional models shows that the heterogeneous model has higher reliability due to the consideration of the appropriate properties for the different regions of the brain. The presented model can contribute to personalized medicine, aid in understanding the dynamics of tumor growth, optimize treatment regimens, and develop adaptive therapy strategies.

Selected aspects of avascular tumor growth reproduced by a hybrid model of cell dynamics and chemical kinetics

We here propose a hybrid computational framework to reproduce and analyze aspects of the avascular progression of a generic solid tumor. Our method first employs an individual-based approach to represent the population of tumor cells, which are distinguished in viable and necrotic agents. The active part of the disease is in turn differentiated according to a set of metabolic states. We then describe the spatio-temporal evolution of the concentration of oxygen and of tumor-secreted proteolytic enzymes using partial differential equations (PDEs). A differential equation finally governs the local degradation of the extracellular matrix (ECM) by the malignant mass. Numerical realizations of the model are run to reproduce tumor growth and invasion in a number scenarios that differ for cell properties (adhesiveness, duplication potential, proteolytic activity) and/or environmental conditions (level of tissue oxygenation and matrix density pattern). In particular, our simulations suggest that tumor aggressiveness, in terms of invasive depth and extension of necrotic tissue, can be reduced by (i) stable cell-cell contact interactions, (ii) poor tendency of malignant agents to chemotactically move upon oxygen gradients, and (iii) presence of an overdense matrix, if coupled by a disrupted proteolytic activity of the disease.

Free and Interfacial Boundaries in Individual-Based Models of Multicellular Biological systems

Coordination of cell behaviour is key to a myriad of biological processes including tissue morphogenesis, wound healing, and tumour growth. As such, individual-based computational models, which explicitly describe inter-cellular interactions, are commonly used to model collective cell dynamics. However, when using individual-based models, it is unclear how descriptions of cell boundaries affect overall population dynamics. In order to investigate this we define three cell boundary descriptions of varying complexities for each of three widely used off-lattice individual-based models: overlapping spheres, Voronoi tessellation, and vertex models. We apply our models to multiple biological scenarios to investigate how cell boundary description can influence tissue-scale behaviour. We find that the Voronoi tessellation model is most sensitive to changes in the cell boundary description with basic models being inappropriate in many cases. The timescale of tissue evolution when using an overlapping spheres model is coupled to the boundary description. The vertex model is demonstrated to be the most stable to changes in boundary description, though still exhibits timescale sensitivity. When using individual-based computational models one should carefully consider how cell boundaries are defined. To inform future work, we provide an exploration of common individual-based models and cell boundary descriptions in frequently studied biological scenarios and discuss their benefits and disadvantages.

Rules of Engagement: A Guide to Developing Agent-Based Models

Agent-based models (ABM), also called individual-based models, first appeared several decades ago with the promise of nearly real-time simulations of active, autonomous individuals such as animals or objects. The goal of ABMs is to represent a population of individuals (agents) interacting with one another and their environment. Because of their flexible framework, ABMs have been widely applied to study systems in engineering, economics, ecology, and biology. This chapter is intended to guide the users in the development of an agent-based model by discussing conceptual issues, implementation, and pitfalls of ABMs from first principles. As a case study, we consider an ABM of the multi-scale dynamics of cellular interactions in a microbial community. We develop a lattice-free agent-based model of individual cells whose actions of growth, movement, and division are influenced by both their individual processes (cell cycle) and their contact with other cells (adhesion and contact inhibition).

A rigid body framework for multicellular modeling

Off-lattice models are a well-established approach in multicellular modeling, where cells are represented as points that are free to move in space. The representation of cells as point objects is useful in a wide range of settings, particularly when large populations are involved; however, a purely point-based representation is not naturally equipped to deal with objects that have length, such as cell boundaries or external membranes. Here we introduce an off-lattice modeling framework that exploits rigid body mechanics to represent objects using a collection of conjoined one-dimensional edges in a viscosity-dominated system. This framework can be used to represent cells as free moving polygons, to allow epithelial layers to smoothly interact with themselves, to model rod-shaped cells such as bacteria and to robustly represent membranes. We demonstrate that this approach offers solutions to the problems that limit the scope of current off-lattice multicellular models.

On Systems of Active Particles Perturbed by Symmetric Bounded Noises: A Multiscale Kinetic Approach

We consider an ensemble of active particles, i.e., of agents endowed by internal variables u(t). Namely, we assume that the nonlinear dynamics of u is perturbed by realistic bounded symmetric stochastic perturbations acting nonlinearly or linearly. In the absence of birth, death and interactions of the agents (BDIA) the system evolution is ruled by a multidimensional Hypo-Elliptical Fokker–Plank Equation (HEFPE). In presence of nonlocal BDIA, the resulting family of models is thus a Partial Integro-differential Equation with hypo-elliptical terms. In the numerical simulations we focus on a simple case where the unperturbed dynamics of the agents is of logistic type and the bounded perturbations are of the Doering–Cai–Lin noise or the Arctan bounded noise. We then find the evolution and the steady state of the HEFPE. The steady state density is, in some cases, multimodal due to noise-induced transitions. Then we assume the steady state density as the initial condition for the full system evolution. Namely we modeled the vital dynamics of the agents as logistic nonlocal, as it depends on the whole size of the population. Our simulations suggest that both the steady states density and the total population size strongly depends on the type of bounded noise. Phenomena as transitions to bimodality and to asymmetry also occur.

Maintaining the proliferative cell niche in multicellular models of epithelia

The maintenance of the proliferative cell niche is critical to epithelial tissue morphology and function. In this paper we investigate how current modelling methods can result in the erroneous loss of proliferative cells from the proliferative cell niche. Using an established model of the inter-follicular epidermis we find there is a limit to the proliferative cell densities that can be maintained in the basal layer (the niche) if we do not include additional mechanisms to stop the loss of proliferative cells from the niche. We suggest a new methodology that enables maintenance of a desired homeostatic population of proliferative cells in the niche: a rotational force is applied to the two daughter cells during the mitotic phase of division to enforce a particular division direction. We demonstrate that this new methodology achieves this goal. This methodology reflects the regulation of the orientation of cell division.

CancerInSilico: An R/Bioconductor package for combining mathematical and statistical modeling to simulate time course bulk and single cell gene expression data in cancer

Bioinformatics techniques to analyze time course bulk and single cell omics data are advancing. The absence of a known ground truth of the dynamics of molecular changes challenges benchmarking their performance on real data. Realistic simulated time-course datasets are essential to assess the performance of time course bioinformatics algorithms. We develop an R/Bioconductor package, CancerInSilico, to simulate bulk and single cell transcriptional data from a known ground truth obtained from mathematical models of cellular systems. This package contains a general R infrastructure for running cell-based models and simulating gene expression data based on the model states. We show how to use this package to simulate a gene expression data set and consequently benchmark analysis methods on this data set with a known ground truth. The package is freely available via Bioconductor: http://bioconductor.org/packages/CancerInSilico/

An agent-based model for drug-radiation interactions in the tumour microenvironment: Hypoxia-activated prodrug SN30000 in multicellular tumour spheroids

Multicellular tumour spheroids capture many characteristics of human tumour microenvironments, including hypoxia, and represent an experimentally tractable in vitro model for studying interactions between radiotherapy and anticancer drugs. However, interpreting spheroid data is challenging because of limited ability to observe cell fate within spheroids dynamically. To overcome this limitation, we have developed a hybrid continuum/agent-based model (ABM) for HCT116 tumour spheroids, parameterised using experimental models (monolayers and multilayers) in which reaction and diffusion can be measured directly. In the ABM, cell fate is simulated as a function of local oxygen, glucose and drug concentrations, determined by solving diffusion equations and intracellular reactions. The model is lattice-based, with cells occupying discrete locations on a 3D grid embedded within a coarser grid that encompasses the culture medium; separate solvers are employed for each grid. The generated concentration fields account for depletion in the medium and specify concentration-time profiles within the spheroid. Cell growth and survival are determined by intracellular oxygen and glucose concentrations, the latter based on direct measurement of glucose diffusion/reaction (in multilayers) for the first time. The ABM reproduces known features of spheroids including overall growth rate, its oxygen and glucose dependence, peripheral cell proliferation, central hypoxia and necrosis. We extended the ABM to describe in detail the hypoxia-dependent interaction between ionising radiation and a hypoxia-activated prodrug (SN30000), again using experimentally determined parameters; the model accurately simulated clonogenic cell killing in spheroids, while inclusion of reversible cell cycle delay was required to account for the marked spheroid growth delay after combined radiation and SN30000. This ABM of spheroid growth and response exemplifies the utility of integrating computational and experimental tools for investigating radiation/drug interactions, and highlights the critical importance of understanding oxygen, glucose and drug concentration gradients in interpreting activity of therapeutic agents in spheroid models.

Studies in 3D cultures, notably multicellular tumour spheroids that mimic many features of solid tumours, have great potential for speeding up anticancer drug development. However the increased complexity of 3D cultures makes interpretation of experiments more difficult. We have developed a hybrid continuum/agent-based mathematical model, validated by experiments, to aid interpretation of spheroid experiments in developing drugs designed to eliminate radiation-resistant hypoxic cells. This model includes key features of the tumour microenvironment including oxygen and glucose transport and regions of hypoxia where the cells are resistant to radiation, but sensitive to hypoxia-activated prodrugs such as SN30000. This enables us to predict the growth and cell response in untreated spheroids and compare the results to spheroids treated with radiation and SN30000. We demonstrate good prediction of cellular responses in spheroids treated with radiation and SN30000 and good agreement with spheroid regrowth after treatment when additional effects of cellular growth delay are added. This demonstrates that the modelling approach has potential to improve interpretation of experimental investigations of drug and radiation combinations.

Mathematical modeling of the proliferation gradient in multicellular tumor spheroids

MultiCellular Tumor Spheroids are 3D cell cultures that can accurately reproduce the behavior of solid tumors. It has been experimentally observed that large spheroids exhibit a decreasing gradient of proliferation from the periphery to the center of these multicellular 3D models: the proportion of proliferating cells is higher in the periphery while the non-proliferating quiescent cells increase in depth. In this paper, we propose to investigate the key mechanisms involved in the establishment of this gradient with a Partial Differential Equations model that mimics the experimental set-up of growing spheroids under different nutrients supply conditions. The model consists of mass balance equations on the two cell populations observed in the data: the proliferating cells and the quiescent cells. The spherical symmetry is used to rewrite the model in radial and relative coordinates. Thanks to a rigorous data postprocessing the model is then fit and compared quantitatively with the experimental quantification of the percentage of proliferating cells from EdU immunodetection on 2D spheroid cryosection images. The results of this calibration show that the proliferation gradient observed in spheroids can be quantitatively reproduced by our model.

Mathematical modeling and optimal control problems in brain tumor targeted drug delivery strategies

In this paper, we present a mathematical model that describes tumor-normal cells interaction dynamics focusing on role of drugs in treatment of brain tumors. The goal is to predict distribution and necessary quantity of drugs delivered in drug-therapy by using optimal control framework. The model describes interactions of tumor and normal cells using a system of reactions–diffusion equations involving the drug concentration, tumor cells and normal tissues. The control estimates simultaneously blood perfusion rate, reabsorption rate of drug and drug dosage administered, which affect the effects of brain tumor chemotherapy. First, we develop mathematical framework which models the competition between tumor and normal cells under chemotherapy constraints. Then, existence, uniqueness and regularity of solution of state equations are proved as well as stability results. Afterwards, optimal control problems are formulated in order to minimize the drug delivery and tumor cell burden in different situations. We show existence and uniqueness of optimal solution, and we derive necessary conditions for optimality. Finally, to solve numerically optimal control and optimization problems, we propose and investigate an adjoint multiple-relaxation-time lattice Boltzmann method for a general nonlinear coupled anisotropic convection–diffusion system (which includes the developed model for brain tumor targeted drug delivery system).

Comparing individual-based approaches to modelling the self-organization of multicellular tissues

The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.

A Computational Model for Investigating Tumor Apoptosis Induced by Mesenchymal Stem Cell-Derived Secretome

Apoptosis is a programmed cell death that occurs naturally in physiological and pathological conditions. Defective apoptosis can trigger the development and progression of cancer. Experiments suggest the ability of secretome derived from mesenchymal stem cells (MSC) to induce apoptosis in cancer cells. We develop a hybrid discrete-continuous multiscale model to further investigate the effect of MSC-derived secretome in tumor growth. The model encompasses three biological scales. At the molecular scale, a system of ordinary differential equations regulate the expression of proteins involved in apoptosis signaling pathways. At the cellular scale, discrete equations control cellular migration, phenotypic switching, and proliferation. At the extracellular scale, a system of partial differential equations are employed to describe the dynamics of microenvironmental chemicals concentrations. The simulation is able to produce both avascular tumor growth rate and phenotypic patterns as observed in the experiments. In addition, we obtain good quantitative agreements with the experimental data on the apoptosis of HeLa cancer cells treated with MSC-derived secretome. We use this model to predict the growth of avascular tumor under various secretome concentrations over time.

Fabrication of Biomimetic Bone Tissue Using Mesenchymal Stem Cell-Derived Three-Dimensional Constructs Incorporating Endothelial Cells

The development of technologies to promote vascularization of engineered tissue would drive major developments in tissue engineering and regenerative medicine. Recently, we succeeded in fabricating three-dimensional (3D) cell constructs composed of mesenchymal stem cells (MSCs). However, the majority of cells within the constructs underwent necrosis due to a lack of nutrients and oxygen. We hypothesized that incorporation of vascular endothelial cells would improve the cell survival rate and aid in the fabrication of biomimetic bone tissues in vitro. The purpose of this study was to assess the impact of endothelial cells combined with the MSC constructs (MSC/HUVEC constructs) during short- and long-term culture. When human umbilical vein endothelial cells (HUVECs) were incorporated into the cell constructs, cell viability and growth factor production were increased after 7 days. Furthermore, HUVECs were observed to proliferate and self-organize into reticulate porous structures by interacting with the MSCs. After long-term culture, MSC/HUVEC constructs formed abundant mineralized matrices compared with those composed of MSCs alone. Transmission electron microscopy and qualitative analysis revealed that the mineralized matrices comprised porous cancellous bone-like tissues. These results demonstrate that highly biomimetic bone tissue can be fabricated in vitro by 3D MSC constructs incorporated with HUVECs.

A non-phenomenological model of competition and cooperation to explain population growth behaviors

This paper is an extension of a previous work which proposes a non-phenomenological model of population growth that is based on the interactions among the individuals of a population. In addition to what had already been studied—that the individuals interact competitively—in the present work it is also considered that the individuals interact cooperatively. As a consequence of this new consideration, a richer dynamics is observed. For instance, besides getting the population models already reached from the original version of the model (as the Malthus, Verhulst, Gompertz, Richards, Bertalanffy and power-law growth models), the new formulation also reaches the von Foerster growth model and also a regime of divergence of the population at a finite time. An agent-based model is also presented in order to give support to the analytical results. Moreover, this new approach of the model explains the Allee effect as an emergent behavior of the cooperative and competitive interactions among the individuals. The Allee effect is the characteristic of some populations of increasing the population growth rate in a small-sized population. Whereas the models presented in the literature explain the Allee effect with phenomenological ideas, the model presented here explains this effect by the interactions between the individuals. The model is tested with empirical data to justify its formulation. Another interesting macroscopic emergent behavior from the model proposed is the observation of a regime of population divergence at a finite time. It is interesting that this characteristic is observed in humanity's global population growth. It is shown that in a regime of cooperation, the model fits very well to the human population growth data from 1000 AD to nowadays.

Large-scale parameter studies of cell-based models of tissue morphogenesis using CompuCell3D or VirtualLeaf

Computational, cell-based models, such as the cellular Potts model (CPM), have become a widely used tool to study tissue formation. Most cell-based models mimic the physical properties of cells and their dynamic behavior, and generate images of the tissue that the cells form due to their collective behavior. Due to these intuitive parameters and output, cell-based models are often evaluated visually and the parameters are fine-tuned by hand. To get better insight into how in a cell-based model the microscopic scale (e.g., cell behavior, secreted molecular signals, and cell-ECM interactions) determines the macroscopic scale, we need to generate morphospaces and perform parameter sweeps, involving large numbers of individual simulations. This chapter describes a protocol and presents a set of scripts for automatically setting up, running, and evaluating large-scale parameter sweeps of cell-based models. We demonstrate the use of the protocol using a recent cellular Potts model of blood vessel formation model implemented in CompuCell3D. We show the versatility of the protocol by adapting it to an alternative cell-based modeling framework, VirtualLeaf.

A multilayer grow-or-go model for GBM: effects of invasive cells and anti-angiogenesis on growth

The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblastoma multiforme (GBM) has uncovered unusual tumor responses. Here, we derive a new mathematical model that takes into account the ability of proliferative cells to become invasive under hypoxic conditions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necrosis. The model is able to replicate and justify the clinical observation of rebound growth when AA therapy is discontinued in some patients. The model is interrogated to derive fundamental insights int cancer biology and on the clinical and biological effects of AA drugs. Invasive cells promote tumor growth, which in the long run exceeds the effects of angiogenesis alone. Furthermore, AA drugs increase the fraction of invasive cells in the tumor, which explain progression by fluid-attenuated inversion recovery (FLAIR) signal and the rebound tumor growth when AA is discontinued.

The modulatory effect of cell–cell contact on the tumourigenic potential of pre-malignant epithelial cells: a computational exploration

Malignant development cannot be attributed alone to genetic changes in a single cell, but occurs as a result of the complex interplay between the failure of cellular regulation mechanisms and the presence of a permissive microenvironment. Although E-cadherin is classified as a 'metastasis suppressor' owing to its role in intercellular adhesion, the observation that it may be downregulated at a premalignant stage is indicative of additional roles in neoplastic development. We have used an agent-based computational model to explore the emergent behaviour resulting from the interaction of single and subpopulations of E-cadherin-compromised cells with unaffected normal epithelial cells within a monolayer environment. We have extended this to investigate the importance of local tissue perturbations in the form of scratch-wounding, or ablation of randomly-dispersed normal cells, on the growth of a single cell exhibiting E-cadherin loss. Our results suggest that the microenvironment with respect to localized cell density and normal/E-cadherin-compromised neighbours is crucial in determining whether an abnormal individual cell proliferates or remains dormant within the monolayer. These predictions raise important questions relating to the propensity for individual mutations to give rise to disease, and future experimental exploration of these will enhance our understanding of a complex, multifactorial pathological process.

Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): from microscopic measurements to macroscopic predictions of clinical progression

Ductal carcinoma in situ (DCIS)--a significant precursor to invasive breast cancer--is typically diagnosed as microcalcifications in mammograms. However, the effective use of mammograms and other patient data to plan treatment has been restricted by our limited understanding of DCIS growth and calcification. We develop a mechanistic, agent-based cell model and apply it to DCIS. Cell motion is determined by a balance of biomechanical forces. We use potential functions to model interactions with the basement membrane and amongst cells of unequal size and phenotype. Each cell's phenotype is determined by genomic/proteomic- and microenvironment-dependent stochastic processes. Detailed "sub-models" describe cell volume changes during proliferation and necrosis; we are the first to account for cell calcification. We introduce the first patient-specific calibration method to fully constrain the model based upon clinically-accessible histopathology data. After simulating 45 days of solid-type DCIS with comedonecrosis, the model predicts: necrotic cell lysis acts as a biomechanical stress relief and is responsible for the linear DCIS growth observed in mammography; the rate of DCIS advance varies with the duct radius; the tumour grows 7-10mm per year--consistent with mammographic data; and the mammographic and (post-operative) pathologic sizes are linearly correlated--in quantitative agreement with the clinical literature. Patient histopathology matches the predicted DCIS microstructure: an outer proliferative rim surrounds a stratified necrotic core with nuclear debris on its outer edge and calcification in the centre. This work illustrates that computational modelling can provide new insight on the biophysical underpinnings of cancer. It may 1-day be possible to augment a patient's mammography and other imaging with rigorously-calibrated models that help select optimal surgical margins based upon the patient's histopathologic data.

COMPUTATIONAL CHALLENGES OF TUMOR SPHEROID MODELING

The speed and the versatility of today's computers open up new opportunities to simulate complex biological systems. Here we review a computational approach recently proposed by us to model large tumor cell populations and spheroids, and we put forward general considerations that apply to any fine-grained numerical model of tumors. We discuss ways to bypass computational limitations and discuss our incremental approach, where each step is validated by experimental observations on a quantitative basis. We present a few results on the growth of tumor cells in closed and open environments and of tumor spheroids. This study suggests new ways to explore the initial growth phase of solid tumors and to optimize antitumor treatments.

Cell-oriented modeling of angiogenesis

Due to its significant involvement in various physiological and pathological conditions, angiogenesis (the development of new blood vessels from an existing vasculature) represents an important area of the actual biological research and a field in which mathematical modeling proved particularly useful in supporting the experimental work. In this paper, we focus on a specific modeling strategy, known as "cell-centered" approach. This type of mathematical models work at a "mesoscopic scale," assuming the cell as the natural level of abstraction for computational modeling of development. They treat cells phenomenologically, considering their essential behaviors to study how tissue structure and organization emerge from the collective dynamics of multiple cells. The main contributions of the cell-oriented approach to the study of the angiogenic process will be described. From one side, they have generated "basic science understanding" about the process of capillary assembly during development, growth, and pathology. On the other side, models were also developed supporting "applied biomedical research" for the purpose of identifying new therapeutic targets and clinically relevant approaches for either inhibiting or stimulating angiogenesis.

Evolution of tumor invasiveness: the adaptive tumor microenvironment landscape model

Interactions between cancer cells and their microenvironment are crucial for promoting tumor growth and invasiveness. In the tumor adaptive landscape model, hypoxic and acidic microenvironmental conditions reduce the fitness of cancer cells and significantly restrict their proliferation. This selects for enhanced motility as cancer cells may evolve an invasive phenotype if the consequent cell movement is rewarded by proliferation. Here, we used an integrative approach combining a mathematical tumor adaptive landscape model with experimental studies to examine the evolutionary dynamics that promote an invasive cancer phenotype. Computer simulation results hypothesized an explicit coupling of motility and proliferation in cancer cells. The mathematical modeling results were also experimentally examined by selecting Panc-1 cells with enhanced motility on a fibroblast-derived 3-dimensional matrix for cells that move away from the unfavorable metabolic constraints. After multiple rounds of selection, the cells that adapted through increased motility were characterized for their phenotypic properties compared with stationary cells. Microarray and gene depletion studies showed the role of Rho-GDI2 in regulating both cell movement and proliferation. Together, this work illustrates the partnership between evolutionary mathematical modeling and experimental validation as a potentially useful approach to study the complex dynamics of the tumor microenvironment.

An Adaptive Multigrid Algorithm for Simulating Solid Tumor Growth Using Mixture Models

In this paper we give the details of the numerical solution of a three-dimensional multispecies diffuse interface model of tumor growth, which was derived in (Wise et al., J. Theor. Biol. 253 (2008)) and used to study the development of glioma in (Frieboes et al., NeuroImage 37 (2007) and tumor invasion in (Bearer et al., Cancer Research, 69 (2009)) and (Frieboes et al., J. Theor. Biol. 264 (2010)). The model has a thermodynamic basis, is related to recently developed mixture models, and is capable of providing a detailed description of tumor progression. It utilizes a diffuse interface approach, whereby sharp tumor boundaries are replaced by narrow transition layers that arise due to differential adhesive forces among the cell-species. The model consists of fourth-order nonlinear advection-reaction-diffusion equations (of Cahn-Hilliard-type) for the cell-species coupled with reaction-diffusion equations for the substrate components. Numerical solution of the model is challenging because the equations are coupled, highly nonlinear, and numerically stiff. In this paper we describe a fully adaptive, nonlinear multigrid/finite difference method for efficiently solving the equations. We demonstrate the convergence of the algorithm and we present simulations of tumor growth in 2D and 3D that demonstrate the capabilities of the algorithm in accurately and efficiently simulating the progression of tumors with complex morphologies.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Phenomenological modeling of tumor diameter growth based on a mixed effects model

Over the last few years, taking advantage of the linear kinetics of the tumor growth during the steady-state phase, tumor diameter-based rather than tumor volume-based models have been developed for the phenomenological modeling of tumor growth. In this study, we propose a new tumor diameter growth model characterizing early, late and steady-state treatment effects. Model parameters consist of growth rhythms, growth delays and time constants and are meaningful for biologists. Biological experiments provide in vivo longitudinal data. The latter are analyzed using a mixed effects model based on the new diameter growth function, to take into account inter-mouse variability and treatment factors. The relevance of the tumor growth mixed model is firstly assessed by analyzing the effects of three therapeutic strategies for cancer treatment (radiotherapy, concomitant radiochemotherapy and photodynamic therapy) administered on mice. Then, effects of the radiochemotherapy treatment duration are estimated within the mixed model. The results highlight the model suitability for analyzing therapeutic efficiency, comparing treatment responses and optimizing, when used in combination with optimal experiment design, anti-cancer treatment modalities.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

The virtual cell--a candidate co-ordinator for 'middle-out' modelling of biological systems

Understanding the functioning of biological systems depends on tackling complexity spanning spatial scales from genome to organ to whole organism. The basic unit of life, the cell, acts to co-ordinate information received across these scales and processes the myriad of signals to produce an integrated cellular response. Cells interact with and respond to other cells through direct or indirect contact, resulting in emergent structure and function of tissues and organs. Systems biology has traditionally used either a 'top-down' or 'bottom-up' approach. However, neither approach takes account of heterogeneity or 'noise', which is an inherent feature of cellular behaviour and may have significant impact on system level behaviour. We review existing approaches to modelling that use cellular automata or agent-based methodologies, where individual cells are represented as equivalent virtual entities governed by simple rules. These paradigms allow a direct one-to-one mapping between real and virtual cells that can be exploited in terms of acquiring parameters from experimental systems, or for model validation. Such models are inherently extensible and can be integrated with other modelling modalities (e.g. partial or ordinary differential equations) to model multi-scale phenomena. Alternatively, hierarchical agent models may be used to explore the functions of biological systems across temporal and spatial scales. This review examines individual-based models and the application of the paradigm to explore multi-scale phenomena in biology. In so doing, it demonstrates how cellular-based models have begun to play an important role in the development of 'middle-out' models, but with considerable potential for future development.

Multicell simulations of development and disease using the CompuCell3D simulation environment

Mathematical modeling and computer simulation have become crucial to biological fields from genomics to ecology. However, multicell, tissue-level simulations of development and disease have lagged behind other areas because they are mathematically more complex and lack easy-to-use software tools that allow building and running in silico experiments without requiring in-depth knowledge of programming. This tutorial introduces Glazier-Graner-Hogeweg (GGH) multicell simulations and CompuCell3D, a simulation framework that allows users to build, test, and run GGH simulations.

Individual-based and continuum models of growing cell populations: a comparison

In this paper we compare two alternative theoretical approaches for simulating the growth of cell aggregates in vitro: individual cell (agent)-based models and continuum models. We show by a quantitative analysis of both a biophysical agent-based and a continuum mechanical model that for densely packed aggregates the expansion of the cell population is dominated by cell proliferation controlled by mechanical stress. The biophysical agent-based model introduced earlier (Drasdo and Hoehme in Phys Biol 2:133-147, 2005) approximates each cell as an isotropic, homogeneous, elastic, spherical object parameterised by measurable biophysical and cell-biological quantities and has been shown by comparison to experimental findings to explain the growth patterns of dense monolayers and multicellular spheroids. Both models exhibit the same growth kinetics, with initial exponential growth of the population size and aggregate diameter followed by linear growth of the diameter and power-law growth of the cell population size. Very sparse monolayers can be explained by a very small or absent cell-cell adhesion and large random cell migration. In this case the expansion speed is not controlled by mechanical stress but by random cell migration and can be modelled by the Fisher-Kolmogorov-Petrovskii-Piskounov (FKPP) reaction-diffusion equation. The growth kinetics differs from that of densely packed aggregates in that the initial spread, as quantified by the radius of gyration, is diffusive. Since simulations of the lattice-free agent-based model in the case of very large random migration are too long to be practical, lattice-based cellular automaton (CA) models have to be used for a quantitative analysis of sparse monolayers. Analysis of these dense monolayers leads to the identification of a critical parameter of the CA model so that eventually a hierarchy of three model types (a detailed biophysical lattice-free model, a rule-based cellular automaton and a continuum approach) emerge which yield the same growth pattern for dense and sparse cell aggregates.

Multicellular sprouting in vitro

Cell motility and its guidance through cell-cell contacts is instrumental in vasculogenesis and in other developmental or pathological processes as well. During vasculogenesis, multicellular sprouts invade rapidly into avascular areas, eventually creating a polygonal pattern. Sprout elongation, in turn, depends on a continuous supply of endothelial cells, streaming along the sprout toward its tip. As long-term videomicroscopy of in vitro cell cultures reveal, cell lines such as C6 gliomas or 3T3 fibroblasts form multicellular linear arrangements in vitro, similar to the multicellular vasculogenic sprouts. We show evidence that close contact with elongated cells enhances and guides cell motility. To model the patterning process we augmented the widely used cellular Potts model with an inherently nonequilibrium interaction whereby surfaces of elongated cells become more preferred adhesion substrates than surfaces of well-spread, isotropic cells.

Integrative mathematical oncology

Cancer research attracts broad resources and scientists from many disciplines, and has produced some impressive advances in the treatment and understanding of this disease. However, a comprehensive mechanistic view of the cancer process remains elusive. To achieve this it seems clear that one must assemble a physically integrated team of interdisciplinary scientists that includes mathematicians, to develop mathematical models of tumorigenesis as a complex process. Examining these models and validating their findings by experimental and clinical observations seems to be one way to reconcile molecular reductionist with quantitative holistic approaches and produce an integrative mathematical oncology view of cancer progression.

Grid-free models of multicellular systems, with an application to large-scale vortices accompanying primitive streak formation

This paper is comprised of two parts. In the first we provide a brief overview of grid-free methods for modeling multicellular systems. We focus on an approach based on Langevin equations, in which inertia is ignored, and stochastic effects on cell motion are included. The discussion starts with simpler models, in which cells are modeled as adhesive spheres. We then turn to more sophisticated approaches in which nontrivial cell shape is accommodated, including the recently introduced Subcellular Element Model, in which each cell is described as a cluster of adhesively coupled over-damped subcellular elements, representing patches of cytoskeleton. In the second part of the paper we illustrate the use of a standard grid-free cell-based model to computationally probe interesting new features associated with primitive streak formation in the chick embryo. Streak formation is a key developmental step in amniotes (i.e., birds, reptiles, and mammals), and can be observed in detail in the chick embryo, where the streak extends across a tightly-packed two-dimensional sheet (the epiblast) comprised of about 50,000 cells. The Weijer group [Cui, Yang, Chuai, Glazier, and Weijer, Dev. Biol. 284 (2005) 37-47] recently observed that streak formation is accompanied by coordinated cell movement lateral to the streak, resulting in two large counter-rotating vortices. We study a mechanism based on cell polarity (in the plane of the epiblast) that provides an explanation for these vortices, and test it successfully using computer simulations. This mechanism is robust, since the emergent vortex formation depends only on the gross features of the initial spatial distribution of planar polarity in the epiblast.

Can tissue surface tension drive somite formation?

The prevailing model of somitogenesis supposes that the presomitic mesoderm is segmented into somites by a clock and wavefront mechanism. During segmentation, mesenchymal cells undergo compaction, followed by a detachment of the presumptive somite from the rest of the presomitic mesoderm and the subsequent morphological changes leading to rounded somites. We investigate the possibility that minimization of tissue surface tension drives the somite sculpting processes. Given the time in which somite formation occurs and the high bulk viscosities of tissues, we find that only small changes in shape and form of tissue typically occur through cell movement driven by tissue surface tension. This is particularly true for somitogenesis in the zebrafish. Hence it is unlikely that such processes are the sole and major driving force behind somite formation. We propose a simple chemotactic mechanism that together with heightened adhesion can account for the morphological changes in the time allotted for somite formation.

An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development

A biomechanical approach in modelling the growth and division of a single fully deformable cell by using an immersed boundary method with distributed sources is presented, and its application to model the early tumour development is discussed. This mathematical technique couples a continuous description of a viscous incompressible cytoplasm with the dynamics of separate elastic cells, containing their own point nuclei, elastic plasma membranes with membrane receptors, and individually regulated cell processes. This model enables one to focus on the biomechanical properties of individual cells and on communication between cells and their microenvironment, simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue. Several examples of early tumours growing in various geometrical configurations and with distinct conditions of their initiation and progression are also presented to show the strength of our approach in modelling different topologies of the growing tissues in distinct biochemical conditions of the surrounding media.

Macrophage-based anti-cancer therapy: modelling different modes of tumour targeting

Tumour hypoxia is associated with poor drug delivery and low rates of cell proliferation, factors that limit the efficacy of therapies that target proliferating cells. Since macrophages localise within hypoxic regions, a promising way to target hypoxic tumour cells involves engineering macrophages to express therapeutic genes under hypoxia. In this paper we develop mathematical models to compare the responses of avascular tumour spheroids to two modes of action: either the macrophages deliver an enzyme that activates an externally applied prodrug (bystander model), or they deliver cytotoxic factors directly (local model). The models we develop comprise partial differential equations for a multiphase mixture of tumour cells, macrophages and extracellular fluid, coupled to a moving boundary representing the spheroid surface. Chemical constituents, such as oxygen and drugs, diffuse within the multiphase mixture. Simulations of both models show the spheroid evolving to an equilibrium or to a travelling wave (multiple stable solutions are also possible). We uncover the parameter dependence of the wave speed and steady-state tumour size, and bifurcations between these solution forms. For some parameter sets, adding extra macrophages has a counterintuitive deleterious effect, triggering a bifurcation from bounded to unbounded tumour growth. While these features are common to the bystander and local models, the crucial difference is where cell death occurs. The bystander model is comparable to traditional chemotherapy, with poor targeting of hypoxic tumour cells; however, the local mode of action is more selective for hypoxic regions. We conclude that effective targeting of hypoxic tumour cells may require the use of drugs with limited mobility or whose action does not depend on cell proliferation.

Individual cell-based models of tumor-environment interactions: Multiple effects of CD97 on tumor invasion

The presence of scattered tumor cells at the invading front of several carcinomas has clinical significance. These cells differ in their protein expression from cells in central tumor regions as recently shown for the EGF-TM7 receptor CD97. To understand the impact of such heterogeneity on tumor invasion, we investigated tumor cells with modified CD97 expression in vitro and in vivo. Applying an individual cell-based computer model approach, we linked specific cell properties of these cells to tumor invasion characteristics. CD97 overexpression promoted tumor growth in scid mice, stimulated single cell motility, increased proteolytic activity of matrix metalloproteinases, and secretion of chemokines in vitro in an isoform-specific manner. We demonstrated by computer simulation studies that these effects of CD97 can increase the invasion capacity of tumors. Furthermore, they can cause the appearance of scattered tumor cells at the invasion front. We identified local tumor environment interactions as triggers of these multiple capabilities. Experimentally, our simulation results are supported by the finding that CD97 expression in tumor cells is regulated by their environment. Our combined experimental-theoretical analysis provides novel insight to how variations of individual cell properties can be linked to individual patterns of tumor cell invasion.

Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics

Background:

One of the mechanisms that ensure cancer robustness is tumor heterogeneity, and its effects on tumor cells dynamics have to be taken into account when studying cancer progression. There is no unifying theoretical framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity.

Results:

Here we formulate a modeling approach that naturally takes stock of inherent cancer cell heterogeneity and illustrate it with a model of interaction between a tumor and an oncolytic virus. We show that several phenomena that are absent in homogeneous models, such as cancer recurrence, tumor dormancy, and others, appear in heterogeneous setting. We also demonstrate that, within the applied modeling framework, to overcome the adverse effect of tumor cell heterogeneity on the outcome of cancer treatment, a heterogeneous population of an oncolytic virus must be used. Heterogeneity in parameters of the model, such as tumor cell susceptibility to virus infection and the ability of an oncolytic virus to infect tumor cells, can lead to complex, irregular evolution of the tumor. Thus, quasi-chaotic behavior of the tumor-virus system can be caused not only by random perturbations but also by the heterogeneity of the tumor and the virus.

Conclusion:

The modeling approach described here reveals the importance of tumor cell and virus heterogeneity for the outcome of cancer therapy. It should be straightforward to apply these techniques to mathematical modeling of other types of anticancer therapy.

Reviewers:

Leonid Hanin (nominated by Arcady Mushegian), Natalia Komarova (nominated by Orly Alter), and David Krakauer.

Individual cell-based models of the spatial-temporal organization of multicellular systems--achievements and limitations

Computational approaches of multicellular assemblies have reached a stage where they may contribute to unveil the processes that underlie the organization of tissues and multicellular aggregates. In this article, we briefly review and present some new results on a number of 3D lattice free individual cell-based mathematical models of epithelial cell populations. The models we consider here are parameterized by bio-physical and cell-biological quantities on the level of an individual cell. Eventually, they aim at predicting the dynamics of the biological processes on the tissue level. We focus on a number of systems, the growth of cell populations in vitro, and the spatial-temporal organization of regenerative tissues. For selected examples we compare different model approaches and show that the qualitative results are robust with respect to many model details. Hence, for the qualitative features and largely for the quantitative features many model details do not matter as long as characteristic biological features and mechanisms are correctly represented. For a quantitative prediction, the control of the bio-physical and cell-biological parameters on the molecular scale has to be known. At this point, slide-based cytometry may contribute. It permits to track the fate of cells and other tissue subunits in time and validated the organization processes predicted by the mathematical models.

Cell elongation is key to in silico replication of in vitro vasculogenesis and subsequent remodeling

Vasculogenesis, the de novo growth of the primary vascular network from initially dispersed endothelial cells, is the first step in the development of the circulatory system in vertebrates. In the first stages of vasculogenesis, endothelial cells elongate and form a network-like structure, called the primary capillary plexus, which subsequently remodels, with the size of the vacancies between ribbons of endothelial cells coarsening over time. To isolate such intrinsic morphogenetic ability of endothelial cells from its regulation by long-range guidance cues and additional cell types, we use an in vitro model of human umbilical vein endothelial cells (HUVEC) in Matrigel. This quasi-two-dimensional endothelial cell culture model would most closely correspond to vasculogenesis in flat areas of the embryo like the yolk sac. Several studies have used continuum mathematical models to explore in vitro vasculogenesis: such models describe cell ensembles but ignore the endothelial cells' shapes and active surface fluctuations. While these models initially reproduce vascular-like morphologies, they eventually stabilize into a disconnected pattern of vascular "islands." Also, they fail to reproduce temporally correct network coarsening. Using a cell-centered computational model, we show that the endothelial cells' elongated shape is key to correct spatiotemporal in silico replication of stable vascular network growth. We validate our simulation results against HUVEC cultures using time-resolved image analysis and find that our simulations quantitatively reproduce in vitro vasculogenesis and subsequent in vitro remodeling.

Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model

We study multicellular tumor spheroids by introducing a new three-dimensional agent-based Voronoi-Delaunay hybrid model. In this model, the cell shape varies from spherical in thin solution to convex polyhedral in dense tissues. The next neighbors of the cells are provided by a weighted Delaunay triangulation with on average linear computational complexity. The cellular interactions include direct elastic forces and cell-cell as well as cell-matrix adhesion. The spatiotemporal distribution of two nutrients--oxygen and glucose--is described by reaction-diffusion equations. Viable cells consume the nutrients, which are converted into biomass by increasing the cell size during the G1 phase. We test hypotheses on the functional dependence of the uptake rates and use computer simulations to find suitable mechanisms for the induction of necrosis. This is done by comparing the outcome with experimental growth curves, where the best fit leads to an unexpected ratio of oxygen and glucose uptake rates. The model relies on physical quantities and can easily be generalized towards tissues involving different cell types. In addition, it provides many features that can be directly compared with the experiment.